The question is from the relative speed . It is a straightforward question and tests the understanding of the concepts. Time Speed and Distance is a favorite in CAT Exam, and appears more often than expected in the CAT Quantitative Aptitude section in the CAT Exam

Question 24: Two persons A and B start moving at each other from point P and Q respectively which are 1400 Km apart. Speed of A is 50 Km/hr and that of B is 20 Km/hr. How far is A from Q when he meets B for the 22nd time?

- 1000 km
- 400 km
- 800 km
- 1400 km

800 km

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Initially A starts from P and B starts from Q.

The ratio of Speeds of A and B is 5 : 2.

That means, in the interval A completes 5 laps between P and Q, B completes 2 laps between Q and P.

At the end of this interval A and B both are at Q. ( Because B starts from Q and runs even number of laps, so he ends in Q, and A starts from P and runs odd number of laps, so he ends on the other side which is Q)

Concentrating on A alone, During these 5 laps he meets B exactly once per lap somewhere between P and Q. (For the 5th time he meets at Q)

Let's consider that A runs another 5 laps(or B runs 2 laps). At the end of which A reaches P and B remains at Q.

Will they meet 5 times again??

Observe that A and B are starting from the same point Q. So during the first lap (of the second set of 5 laps) A and B will not meet each other.

So, A and B meet only for 4 times during the second set of 5 laps.

Consolidating everything observed till now, After A runs 10 laps between P and Q, A and B meet for 9 times (5+4) and end up at their original positions, which is A at P and B at Q.

Since they have reached their initial positions, this becomes cyclic.

So after 20 laps of A, They would be at their original positions, and would have met exactly 18 times (9+9).

So, After meeting for 18 times, they remain at their original positions.

But we are concerned with their positions after the 22nd meet.

Since 22 = 18 + 4, finding the location of the 4th meet from the original positions, gives the location of the 22nd meet.

We can do it in two ways.

1) Calculate the 1st 2nd 3rd and 4th meet iteratively,(which is lengthy and time consuming)

or

2) We know the positions after 5 more laps( Equivalent to 5 more meets), as Both at Q. If we can back track one meet from there. Bingo!.

Let's go by the second way. After 23 meets A and B both are at Q.

Let us say that the 22nd meet happened at a point Y, which is located between P and Q.

Since A and B meet each other, running in the same direction for 23rd time, for 22nd time they must be running in the opposite direction.

That means B travels from Y to Q, While A travels from Y to P to Q.

So together between them they cover 2*1400km.

Since their speeds are in 5:2 ratio,

let's say Y to Q = 2x, which make Y to P to Q = 5x.

And 2x + 5x = 2 * 1400

x = 400

The 22nd meet happened at Y, Distance between Y to Q = 2x = 2*400 = 800km

So, A is 800km away from Q, when he meets B for 22nd time.

The question is **"How far is A from Q when he meets B for the 22nd time?"**

Choice C is the correct answer.

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